For groups and vector spaces, the ‘structure-preserving’ functions are, respectively, group homomorphisms and linear transformations. The analogous functions for FG-modules are called FG-homomorphisms, and we introduce these in this chapter.
Let V and W be FG-modules. A function ϑ: V → W is said to be an FG-homomorphism if ϑ is a linear transformation and
In other words, if ϑ sends v to w then it sends vg to wg.
Note that if G is a finite group and ϑ: V → W is an FG-homomorphism, then for all v ∈ V and , we have
The next result shows that FG-homomorphisms give rise to FG-submodules in ...